The NSGA-II is the most prominent multi-objective evolutionary algorithm (cited more than 50,000 times). Very recently, a mathematical runtime analysis has proven that this algorithm can have enormous difficulties when the number of objectives is larger than two (Zheng, Doerr. IEEE Transactions on Evolutionary Computation (2024)). However, this result was shown only for the OneMinMax benchmark problem, which has the particularity that all solutions are on the Pareto front, a fact heavily exploited in the proof of this result. In this work, we show a comparable result for the LeadingOnesTrailingZeroes benchmark. This popular benchmark problem appears more natural in that most of its solutions are not on the Pareto front. With a careful analysis of the population dynamics of the NGSA-II optimizing this benchmark, we manage to show that when the population grows on the Pareto front, then it does so much faster by creating known Pareto optima than by spreading out on the Pareto front. Consequently, already when still a constant fraction of the Pareto front is unexplored, the crowding distance becomes the crucial selection mechanism, and thus the same problems arise as in the optimization of OneMinMax. With these and some further arguments, we show that the NSGA-II, with a population size by at most a constant factor larger than the Pareto front, cannot compute the Pareto front in less than exponential time.

翻译：NSGA-II是最具代表性的多目标进化算法（引用次数超过50,000次）。近期一项数学运行时分析证明，当目标数量超过两个时，该算法可能面临巨大困难（Zheng, Doerr. IEEE Transactions on Evolutionary Computation (2024)）。然而，该结论仅在OneMinMax基准问题上得到验证——该问题的特殊性在于所有解均位于帕累托前沿，且证明过程深度依赖这一特性。本研究针对LeadingOnesTrailingZeroes基准问题展示了可比性结论。该经典基准问题更具普遍意义，因其大多数解并不位于帕累托前沿。通过对NSGA-II优化该基准问题时种群动态的精细分析，我们发现：当种群在帕累托前沿扩展时，通过生成已知帕累托最优解的方式比沿帕累托前沿扩散的方式快得多。这导致当帕累托前沿仍有常数比例区域未被探索时，拥挤距离机制已成为关键选择依据，从而产生与优化OneMinMax时间样的问题。基于这些论证及进一步分析，我们证明：当种群规模至多为帕累托前沿大小的常数倍时，NSGA-II无法在低于指数级的时间内计算出完整的帕累托前沿。